PHAN RIENG Thi sinh chi dugfc chgn lam mpt trong hai phan phan A hoac B A... Hinh chieu vuong goc ciia dinh S len mat phang ABCD la trong tam cua tam giac BCD.. Tinh theo a the tich kh
Trang 1ruyeii chifit & Gi&i thifu dethi Todn hqc - Nguyen Phii Khdnh , Nguyen Tai Thu
Gpi A la bien co lay duoc 8 vien c6 du ca 3 mau
A la bien co lay dugc 8 vien khong dii ca 3 mau Khi do
THl Lay dugc 8 vien co diing 1 mau (chi xay ra lay dug-c 8 bi vang) v | y co
B Theo chUcrng trinh nang cao
Cau7.b: B = B C n d , => B(0;-1) => B M = (2;2) Do do B M la mgt vec to phap
Ggi H la hinh chieu ciia B len A Ta co: BH < A B , khoang each tir B den
x = l + t
Ion nhat khi H = A, VTCP cua A la u ^ = U j ; A B =(1;-1;-1)=:>A:
Cau 9.b: Xet z = 0 la nghi|m cua phuong trinh Xet z * 0 Dat z = a + bi (a,b e ]R ,a2 + b^ > o), tu gia thiet ta co:
Lay (1) tru (2) ve theove taco 9a2 = 4b2 <::> a^ = - b ^ (3)
The ( 3 ) vao ( l ) ta dugc : ^ b 2 b^ + b « b = 4 a = — , (do a > 0,b > 0)
Trang 2Tuyen chgn b Giai thi?u <fe thi loan hgc - Nguyen nu Khanli, Nguyen lai iniu
DETHiTH(jfs628
I PHAN CHUNG CHO TAT CA CAC T H I S I N H
Cau 1: Cho ham so: y = x-* - 3mx^ + 9x +1 c6 do thi {C^)
a) Khao sat su bien thien va ve do thi (CQ) ciia ham so
b) Gia six duong t h i n g ( d ) : y = x +10 - 3m cat do thj [C^) cua ham so tgi 3
diem phan biet A , B , C c6 hoanh do ian lugt Xi,X2,X3. Tim m de: xf +X2 +x^ <11
Cau 2: Giai phuong trinh : cos-* x + sin'' x = cosx + sin2x + sinx
Cau 3: Tim m de he phuong trinh
Cau 5: Cho hinh chop S.ABCD c6 day ABCD la hinh thang vuong tai A, B Biet
A D = 2AB = 2BC = 2a, SA = SD = SC = 3a Tinh the tich khoi chop SABC va
khoang each giira hai duong thang SB va CD
1 < a , b , c < 4
Cau 6: Cho cac so thuc a,b,c thoa man dieu kien
a + b + 2c = 8' Tim gia trj Ion nhat cua P = a'^ + b^ + 5c^
I I PHAN R I E N G T h i sinh chi dugc chpn lam mpt trong hai phan (phan
A hoac B)
A T h e o chiTofng t r i n h c h u a n
2 2 Cau 7.a: Trong mat ph5ng tc?a do Oxy, cho duong tron ( C ) : ( x - l ) +(y+2) =1
va duong thang (A) : 2x - y +1 = 0 Tim diem A thuoc duong thSng ( A ) sao cho
t u A ke dupe cac tiep tuyen AB, AC (B, C la cac tiep diem) den duong tron (C)
dong thai di^n tich tarn giac ABC bSng 2,7
Cau 8,a: Trong mat ph^ng Oxyz, cho diem M ( l ; 3; 1), duong thang d:
^ ~ ^ = y±l = I va mat phang ( P ) : x - y + 2z + 5 = 0 Viet phuong trinh mat
phang ( Q ) d i qua M , song song voi d va tao v6i ( P ) mot goc cp thoa coscp = ^
Cau 9.a: Cho a, p la hai so phuc lien hgp thoa la so thuc va a + p = 2>/3 Tinh a
B T h e o c h U o r n g t r i n h n a n g c a o
Cau 7.b: Trong mat phSng tpa dp Oxy, cho duong tron ( C ) : x^+y^-2x-4y-4=0
CO tam I va diem M(3;0) Viet phuong trinh duong thang A, biet A cat (C) tai
hai diem phan biet A , B sao cho t u giac A B I M la hinh binh hanh
Cau 8.b: Trong mat phang Oxyz, cho mp ( p ) : x + 2y + z - 3 = 0 , duong
X — 1 y + 1 z — 2
thang A : - — = —— = _ _ va diem A ( 4 ; l ; - 3 ) Viet phuong trinh duon^
thSng d nam trong (P), biet d cat A va khoang each t u A den d bang 42 ,
Cau 9.b: Tim c biet a, b va c la cac so nguyen duong thoa man c=(a+bi)'^ -107i
Hl/dNG DAN GIAI
I PHAN CHUNG CHO TAT CA CAC THI SINH Cau 1:
a) Danh cho ban doc
b) Ham so da cho xac dinh tren R
Phuong trinh hoanh do giao diem ciia (C) voi duong thang (d) la
-3mx^ + 8x + 3 m - 9 = 0 < : > ( x - l ) r x 2 + ( l - 3 m ) x + 9 - 3 m l = 0 ( l )
o X = 1 ( gia su X3 = 1) hoac x^ + ( l - 3m)x + 9 - 3m = 0 (2)
De duong thSng (d) cat (C) tai 3 diem phan bi?t thi phuong trinh ( l ) c6 3
nghiem phan biet <^ phuong trinh (2) c6 2 nghiem phan biet khac 1, tuc la phai c6:
1^ la gia trj can tim
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Trang 3Cau 2: Phuang trinh du<?c bie'n doi duoi d^ng:
(cosx + sinx)(l-cosx.sinx) = cosx + sin2x + sinx
<=> i(cosx + sinx)sin2x=sin2xo (cosx + sinx-2)sin2x=0
<=> sin2x = 0 o 2 x = k7i » x = k ^ ( k e Z )
Vgy, phuong trinh c6 1 ho nghi?m
Cau3:Truvetheove: yjsx^ +x + l-^3x^-x + l = ^jsy^ +y+ -^P/-y+
Xet ham so f (u) = Vsu^ + u+ 1 - Jsu^ - u + l,Vu € #
.f'(u)>0,Vuei^ nen phuong trinh f (x) = f (y) x = y
Khiay taco: N/SX^ + x +1 + Sx^ - x +1 = m, tu day tim duoc m>2
Cau4: t^^^xle"+lnx)
Cau 5: Theo gia thie't ta c6 BC = AB = a
Gpi H la trung diem ciia AD => HA = HD = a
fCH = a
Tu gia thie't => ABCH la hinh vuong canh a tam O =>
Trong tam giac AACDco C H la trung tuyen va C H = i A D : ^ A A C D
vuong tai C H la tam duong tron ngoai tiep tam giac AACD
Ta CO SB va CD la hai duong thSng cheo nhau
CD//(SBH) , , ,
SB c (SBH) ^ '^(^^'SB) = ^[CD'(SBH)] = d[C(SBH) Mat khac <
f CO 1 H B
Ta CO ^ C O l ( S B H ) ^ C O = d C,(SBH)
[ C O I S H ^ ' L 'V /J 2 Cau 6: P = a^ + b^ + 5c^ = (a + hf -3ab(a + b) + 5c^ = (8 - 2c)^ - 3ab(8 -2c) + 5c^
o P = -3c^ + 96c^ - 384c + 512 - 3ab (8 - 2c)
Ta CO ( a - l ) ( b - l ) > 0 = > a b - ( a + b) + l>0=>ab>a + b - l = 8 - 2 c - l = 7-2c ab(8 - 2c) > (7 - 2c)(8 - 2c) => -3ab(8 - 2c) < -3(7 - 2c)(8 - 2c)
Trang 428 + 7VT0 r, u - 28-7VT0
f(l) = 131, f 28-7VTo'l =, f(3) = 137
Vay gia trj Ion nliat cua P la 137, dat dugc khi c = 3,a = l,b = 1
I I PHAN RIENG Thi sinh chi dugfc chgn lam mpt trong hai phan (phan A
hoac B)
A Theo chUorng trinh chuan
Cau 7.a:
Duong tron (C) c6 tarn l ( l ; - 2 ) , ban kinh R = l
Honnua: S^BIC = S A B C + S B , C « I B A B = ilB2sinBIG + iAB2sinBAG (2)
Tie (l) va (2), suy ra:
2IB.AB = (iB^ + AB^ jsin BAG => sin BAG = 2IB.AB
Cau9 a: Gia su a = a + ib ( a , b £ K ) t h i p = a - i b
Trang 5DETHITHljrsd29
I PHAN CHUNG CHO TAT CA CAC THI SINH
Cau 1: Cho ham so y = - ( m ^ + m - sjx + - 3 m + 2 ( l ) , c6 do thi (C^)
a) Khao sat su bien thien va ve do thj ( C Q ) cua ham so
b) Tim tat ca cac gia tri thuc cua m sao cho do thi ham so ( l ) cat duong
th5ng y = 2 tai ba diem phan biet c6 hoanh dp Ian Ixxgt la X i , X 2 , X 3 va dong
thoi thoa man dang thuc X j + X j + X3 = 18
3 X 3 X
sm'^ ~ cos
-2 + sin X - = cosx Cau 2: Giai phuong trinh:
Cau 3: Giai phuong trinh: \lx^-5\ 6 + N / X ^ + Vx + 21 = V ? + 1 9 x -42
2
Cau 4: Tinh tich phan: I - J(x + l)lnx.e''dx
1
Cau 5: Cho hinh chop S.ABCD c6 day ABCD la hinh binh hanh thoa man AB = 2a,
BC = aV2,BD = a%/6 Hinh chieu vuong goc ciia dinh S len mat phang (ABCD)
la trong tam cua tam giac BCD Tinh theo a the tich khoi chop S.ABCD, biet
rang khoang each giua hai duong thSng AC va SB bang a
Cau 6: Cho ba so a,b,c > 0 thoa a + b + c < 1,5 Tim gia trj nho nhat cua bleu
A Theo chuorng trinh chuan
Cau 7.a: Trong mat phSng tQa dg Oxy, cho diem 1(2; 4) va hai duong than;.;
d i : 2 x - y - 2 = 0, d 2 : 2 x + y - 2 = 0
Viet phuong trinh duong tron tam I cat dj t^ii hai diem A, B va cat d 2 t?'
hai diem C, D sao cho AB + CD = — ~ •
o Cau 8.a: Trong khong gian vai h? tQa dp Oxyz, cho mat cau
(S): ( x - l ) ' + ( y - 2 f + ( z - 3 f =9vadudngthang A: ^ ^ ^ ^ ^ ^
190
Viet phuong trinh mat phSng (p) di qua M(4;3;4), song song voi duong
thang A va tie'p xuc voi mat cau (S)
Cau 9.a: Tim so phuc z thoa man phuong trinh z.z + z^ - ^z - 2zj = 10 + 3i
B Theo chUcrng trinh nang cao
Cau 7.b: Trong mat phMng tpa dp Oxy, lap phuong trinh chinh t3c cua elip ( E )
biet no CO mpt dinh va 2 tieu diem cua ( E ) tao thanh mpt tam giac deu va chu viciia hinh chii-nhat CO so cua ( E ) la 12^2+ V 3 J
Cau 8.b: Trong khong gian tpa dp Oxyz, cho hai duong thSng
Tim M thupc ( A , ) va N thupc Aj sao cho M N = 2N/6 va tam giac A M N
a) Danh cho ban dpc
b) Phuong trinh hoanh dp giao diem cua do thi ham so' (l) va duong thioig y = 2
^' x 3 - ( m 2 + m - 3 ) x + m 2 - 3 m + 2 = 2 < : > x 3 - ( m 2 + m - 3 ) x + m 2 - 3 m = 0
o (x - m)(x^ + mx - m + 3) = 0 o X = m hoac x^ + mx - m + 3 = 0 ( 2 )
Do thi ham so ( l ) c^t duong th^ng y = 2 tai 3 diem phan bi?t khi va chi
khi { 2 ) CO hai nghiem phan bi^t khac
Trang 6Cau 5: Gpi H la hinh chieu vuong goc ciia S len mat phling ( A B C D ) , M la trung diem C D va O la tam ciia day A B C D Do A O la trung tuyen ciia tarn giac
I I PHAN RIENG Thi sinh chi dugc chpn lam mpt trong hai phan (phan A hoac B )
A Theo chUorng trinh chuan
Cau 7.a: G ^ i R la ban kinh duong tron can tim va F, G Ian lugt la hinh chieu
vuong goc ciia I tren dj va d j De thay I F = — , I G = ^
5 5
Laico: FB = VR^ - IF^ = JR^TI, G D = TR^TJ^ = ^R^ - ^6 www.facebook.com/groups/TaiLieuOnThiDaiHoc01/
Trang 7Tuife'u chQH & Giori thifu dethi Toiin HQC - Nguyen Phu Khduh , Nguyen Tii't Thu
Theo bai toan: A B + CD = 2(FB + GD) = ^ R
Cau 8.a: Gpi vecto phap tuyen n = (a;b;c) ciia m|t phang ( p )
Mat phang (P) song song v6i duong thang A -3a + 2b + 2c = 0
d(l;(P)) = R o - ^ = = ^ i ^ i _ = 3 < » ( b - 2 c ) ( 2 b - c ) = 0
VlSb^+Sbc + lSc^
Cau 9.a: Tim so' phuc z thoa man phuong trinh z.z + - - 2zj = 10 + 3i
Gpi z = X + yi (x,y e K),ta c6 z = x - yi va z^ = x^ - y^ +2xyi
Do cac dinh tren tryc Ion va Fj,F2 thMng hang nen Fj,F2 cung voi dinh B(0;b)
tren tryc nho tao thanh mpt tam giac deu
» c^ + b^ = 4c2 » b^ = 3c2 = 3(a2 - b^) <::> 3a2 = Ah^
Hinh chii nhat co so c6 chu vi 2(2a + 2b) = 12(2 + N/S) O a + b = 6 + 3\/3
CauS b: M thupc ( A J ) M(a;a;a +1), N thuQC Aj =>N(b;b + 2;-2b)
Tam giac A M N vuong tai A nen A M 1 A N <=> A M A N = 0
b(a2+b2+25) a^ + b^ = 6
DETHITHUfSOSO
PHAN CHUNG CHO TAT CA CAC T H I SINK
au 1: Cho ham so: y x'* - 2mx^ - 3 c6 do thj la (C^)
a) Khao sat sif bie'n thien va ve do thj (C_j) ciia ham so
b) Tim meM de ban kinh duong tron ngoai tiep tam giac c6 cac dinh la 3
"em cue trj ciia do thj ham so (C,^) dat gia trj nho nhat
au 2: Giai phuong trinh : sin3x = cosx.cos2x|tan2 x + tan2x
iu 3: Giai phuong trinh: x^ - 3x - 4 = V x ^ J x ^ - 4x - 2)
au 4: Tinh tich phan: I = |- 1 f 1 x +
ocos XI t a n ^ x - 4 dx
Cau 5: Cho hinh chop S.ABCD c6 day ABCD la nua luc giac deu va AB = BC =
CD = a Hai mat phing (SAC) va (SBD) cung vuong goc voi mat phSng day
195 www.facebook.com/groups/TaiLieuOnThiDaiHoc01/
Trang 8Tuyeit chgn 6- Gi&i thi?u dethi Todn hqc - Nguyen PM Khdnh , Nguyen Tat Thu
( A B C D ) Tinh theo a the ti'ch cua khol chop S A B C D biet r i n g khoang each
giiia hai duong thang A B va S D b i n g
Cau 6: Cho cac so thuc a,b,c thoa man (a + b + c)^ = 2^3^ + b^ + c^ j T i m gia
tri Ion nhat va gia t r i nho nhat cua bieu thuc : Q = -. ; x-j——; r
(a + b + c)(ab + bc + ca)
I I PHAN R I E N G T h i sinh chi dugrc chgn lam mpt trong hai phan (phan A
hoac B )
A Theo chUcTng trinh chuan
Cau 7.a: Trong mat phang toa dp Oxy, cho 2 duong tron ( C j ) : + = 13 va
( C j ) : (x - 6)^ + y^ = 25 Goi A la giao diem cita ( C j ) va (C2) voi y ^ < 0 Viet
phuong trinh duong thSng d i qua A va cat ( C j ) , ( C j ) theo 2 day cung c6 do
dai bang nhau
Cau 8.a: Trong khong gian v6i h^ toa do Oxyz, cho mat cau (S):
(x + l ) ^ + ( y - l ) ^ = 9 va diem A ( 1 ; 0 ; - 2 ) Viet phuong trinh duong thang
A tiep xiic voi mat cau (S) tai A tao voi tryc Ox mot goc a c6 cos a =
i z - ( l + 3i)z 2 Cau 9.a: Tim so phuc z thoa man dieu ki^n — = z
1 + i
B Theo chUtfng trinh nang cao
Cau 7.b: Trong mat p h i n g toa dp Oxy, cho hinh binh hanh A B C D v o i A(1;1) ,
B(4;5) Tam 1 cua hinh binh hanh thupc duong thSng ( d ) : x + y + 3 = 0 Tim
toa do cac dinh C, D biet rang di^n tich hinh binh hanh A B C D bang 9
Cau 8.b: Trong khong gian voi h^ toa do Oxyz, cho hai duong thang
( d i ) : ^ - ^ ^ = Y , ( d 2 ) : ^ = ^ ^ = Y va mat phSng (P) c6 phuong
trinh x + y - 2 z + 3 = 0 Viet phuong trinh duong thSng A song song v o i (P)
va cat d,,d2 Ian luot tai hai diem A , B sao cho A B = \/29
Cau 9.b: Tir cac so 1, 2, 3, 4, 5 c6 the lap dupe bao nhieu so t u nhien c6 nam chu
so, trong do chii so' 3 c6 mat dung ba Ian, cac chir so'con lai c6 mat khong qu^i
mot Ian Trong cac so' t u nhien noi tren, chpn ngau nhien mpt so', t i m xac suat
de so dupe chon chia he't cho 3
Cty TNHH MTV DWH Khang Vift
Phuong trinh cho tuong duong voi sin3x = ^"^^^"^'"^ % s i n 2 x c o s x
<=> sin x cos 2x + sin 2x cos x = cos2x.sin^x
o sinxcos2x = cos2x.sin^x
cosx
cosx sinx = 0 tan X = 1
cosx + sin 2x cosx
X = kTT
« 1
X = — + krt
4
Doi chieu dieu k i f n, ta tha'y x = k:: thoa man
Vay, phuong trinh c6 1 hp nghi^m
Trang 9-Cau 5: Gpi H la giao diem cua AC va BD Do (SAC) va (SBD) cimg vuong goc
voi mat ph^ng ( A B C D ) nen SH vuong goc voi ( A B C D ) Coi K la hinh chieu
vuong goc cua B len duong thang SD
Do A B C D la nua luc giac deu nen AB vuong
goc voi BD, ket hop voi AB vuong goc voi SH
Vay, VABCD = 3SH.SABCD " 3 y - 4 5
Cau 6: Gia thiet suy ra: a^ + + c^ = 2(ab + be + ca) 198
D|it x = a + b, suy ra (a + b)^ + = 4ab + 2cx => (x - c)^ = 4ab < (a + b)^ = x^
=> 0 < c < 2x Khi do: Q = Neu c = 0 thi Q = 0
( x c f
Neu o O , bang each datt = - > ^ thiQ=^*
( t l ) ,3 •
Xet f(t) = ^* ^\ voi t > i , taco f'(t) = O o t = l hoac t = 5 (t + l f 2
II PHAN RIENG Thi sinh chi dugrc chpn lam mpt trong hai phan (phan A hoac B)
A Theo chUcrng trinh chuan
Cau7.a: A (2;-3) la giao diem (C,) va (Cj)
Phuong trinh duong thang A di qua A c6 dang: a(x-2) + b(y + 3) = 0
Duong tron (Cj) c6 tam O(0;0), ban kinh Rj =>/l3
I Duong Iron (Cj) c6 tam l(6;0), ban kinh =5 Theogiathiet, suyra: R ^ - d ^ ( 0 , A ) = R ^ - d ^ ( l A ) =>x + 3y + 7 = 0
Cau 8.a: Mat cau c6 tam l(-l;l;0), ban kinh R = 3
GQ\ = (a;b;c) la vecto chi phuong cua A, tu gia thiet suy ra lA.n = 0
b = 2a - 2c A tao voi tryc Ox mpt goc a c6 cosa =
- = = = = = = o a = - c ho|c a = c
I , i(a + bi)-(l + 3i)(a-bi) , ,
Cau 9.a: Theo gia thiet, ta c6-^ ^ - = a + b
Trang 1045 9
Vav, CO hai so phuc can tim la = 0, z, = — + — i
B Theo chuorng trinh nang cao
Cau7.b:Giasu C(a;b) =>I D o l e d = i > a + b + 8 = 0 ( l )
x - 1 v - l Phuang trinh duong thSng AB = o 4x - 3y - 1 = 0
Cau 9.b: Gpi aja2a3a4a5 la so tu nhien c6 nam chO so, trong do chii' so 3 c6 mat
diing ba Ian, cac chix so con lai c6 mat khong qua mpt Ian voi ai,a2,a3,a4,a5
6 {1; 2; 3; 4; 5]
S3p chix so 3 vao 3 vj tri, c6 C5 = 10 each
Con lai 2 vj tri, 4 chii so Chpn 2 chii so xep vao 2 vj tri do, c6 C4 = 12 each
Vay khong gian mau c6 10.12 -120 phan tir
Co ( l + 5);3;(2 + 4);3
Gpi A bien co: "so dupe chpn chia het cho 3" c6 2 phuong an
2 c h i f s o c o n l a i l a 1,5 c6 C5.2! = 20 so
2 chu so con lai la 2,4 C O C5 2! = 20 so
Vay bien co A c6 40 phan tu
I PHAN C H U N G CHO TAT CA CAC THI SINH
Cau 1: Cho ham so y = — ^ co do thj la (C)
x - 1 ^ '
a) Khao sat sy bien thien va ve do thj (C) ciia ham so'
b) Tim tat ca cac gia trj tham so m de duong thSng d: y = -x + m - 1 cat do thi (C) ham so tai hai diem A , B sao cho tam giac OAB npi tiep trong duong tron CO ban kinh R = 2\/2
Cau 2: Giai phuong trinh: cos 3x tan 5x = sin 7x
X ' ' + v'' = 5x — V Cau 3: Giai hf phuang trinh: \
e
Cau 4: Tinh tich phan: J = J In x - Vx dx
1 Cau 5: Cho hinh chop tu giac S.ABCD co day ABCD la hinh chCr nhat va AB = a,
BC = aVs Mat phang (SAC) va mat phang (SBD) vuong goc voi day, I thupc canh SC sao cho SI = 2CI va thoa man A I vuong goc voi SC Tinh the tich cua
khoi chop S.ABCD theo a
Cau 6: Cho cac so thuc x, y, z thoa man x^ + y^ + z^ = 9 Tim gia trj Ion nhat ciia bieu thuc: P = (9 + 2yz)(y^z^ - 4yz + s]
II PHAN RIENG Thi sinh chi dugc chpn lam mpt trong hai phan (phan A
hoac B)
A Theo chUorng trinh chuan Cau 7.a: Trong mat phang tpa dp Oxy, cho tam giac ABC co dinh A(1;1), true tam H ( - 1 ; 3 ) , tam duong tron ngoai tiep l(3;-3) Xac djnh tpa dp cac dinh
Trang 11B Theo chUcrng trinh nang cao
Cau 7.b: Trong mat phing toa dp Oxyz cho hinh bmh hanh ABCD c6 D(-6;-6)
Duong trung true cua doan DC c6 phuong trinh ( d ) : 2x + 3y + 17 = 0 va
duong phan giac goc BAC c6 phuong trinh (d'): 5x + y - 3 = 0 Xac djnh toa
dp cac dinh con lai cua hinh binh hanh
Cau 8.b: Trong khong gian tpa dp Oxyz, cho bon duong thang
Viet phuong trinh duong thSng A cat dupe ca bon duong th3ng da cho
Cau 9.b: Tim m de phuong trinh: 27" - 32"^^ +15.3" - m = 0 c6 nghi^m - 1 < x < 2
Hl/dTNGDANGlAl
I PHAN CHUNG CHO TAT CA CAC THI SINH
Cau 1:
a) Danh cho ban dpc
b) Phuong trinh hoanh dp giao diem ctia hai do thj
- ^ ^ = - x + m - l <=>g(x) = x^ - ( m - l ) x + m - l = 0 ( l ) vai x^l
Duong thSng d cat do thj tai hai diem phan bif t khi va chi khi phuong
trinh (1) c6 hai nghiem phan biet, khac 1
x2
A = ( m - l ) - 4 ( m - l ) > 0 o m < 1 hoac m > 5 (2)
g ( l ) = 1^0
Vdi (2) thi d cat do thj (C) ciia ham so tai hai diem A { X J ; - X J + m - l ) ,
B ( X 2 ; - X 2 + m - l ) , gpi X j , X2 la cac nghifm ciia (1), ta c6:
Cau 3: H? da cho tuong duong voi:
I Vay, phuong trinh cho CO nghiem la: X = m7t, , X = + ( m , k 6 Z )
Cau 5: Gpi O la giao diem cua A C va BD ^ ( S A C ) n (SBD) = S O , chung minh dupe S O 1 ( A B C D ) , A C = VBA^TBC^ = 2a O A = O C = a
DatSO = h =>SC = ^|S0^+0C^ =h^+a^
i i
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Trang 12V i SI = 2CI nen IC = ^SC =-Vh^ +
3 3 Tarn giac A I C vuong tai I
I I PHAN R I E N G Thi sinh chi dugc chpn lam mpt trong hai phan (phan A
hoac B )
A Theo chUorng trinh chuan
Cau 7.a: Goi D doi xung voi A qua I thi D ( 5 ; - 7 ) va D nSm tren duong tron
( C ) ngoai tiep tam giac A B C : (x - 3)^ + (y + 3)^ = 20
Gpi J.la trung diem cua H D thi J la trung diem ciia B C nen B C : x - y - 4 = 0
Tpa dp hai diem B , C la nghi^m cua h? phuang trinh: • "'"(>'•'• 3)
[ x - y - 4 = 0
Ma X g < XQ nen hai dinh can tim la B ( - 1 ; - 5 ) va C ( 5 ; l )
Cau 8.a: Goi vecto phap tuyen cua mat ph3ng (P) la n = (a;b;c) ^ 0
Do mat ph^ng ( p ) chua A , B nen n 1 B A trong do B A = (2; - 1 ; 2 ) , suy ra
2 a - b + 2c = 0 = > b - 2 a + 2c ( l )
Hon nCra mat phang (P) t^o voi mat phSng (Oxy) mpt goc ip sao cho
coscp = suy ra 5a^ + Sac - 4c^ = 0 (2)
T u ( l ) va (2) suy ra a = -2c hoac a = - c
5
Cty TNHH MTV DWHKhang Vi?t
Cau 9.a: Dieu kien
Vay, modun cua so phuc z la 3 hay Vs
B Theo chUorng trinh nang cao
Cau 7.b: Phuong trinh DC qua D va vuong goc (d) la: 3x - 2y + 6 = 0
Giao diem ciia DC va (d) la: M ( - 4 ; - 3 ) va cung la trung diem DC
Suy ra tpa dp C ( - 2 ; 0 ) Gpi C la diem doi xung cua C qua d ' thi C ' G A B , phuong trinh C C :
/ I 1^
x - 5 y + 2 - 0 Giao diem C C va d ' la I
U' 2 .Suy ra tpa dp C ( 3 ; l ) Phuong trinh A B qua C vuong goc (d) la: 3x - 2y - 7 = 0
Cau 8 b: dj di qua M(1;2;0)C6 vecto chi phuong i j j = ( l ; 2 ; - 2 )
d j d i q u a N (2; 2; 0)c6 vecto chi phuong Uj = (2;4;-4) = 2iaj d o d o d i / / d 2 Gpi (P) la mat phSng chua dj va d j thi (P) d i qua diem M ( l ; 2 ; 0 ) va c6
Vecto phap tuyen n = M N ; Qj = (O; 2; 2), do do c6 phuong trinh y + z - 2 = 0
Gpi A = d3 n (p) thi tpa dp A thoa h?:
Trang 13Tuye'n chqn €t Gi&i thifu dethi Todn I I Q C - Nguyen Phti Khdttlt, Ngtii/en Tii't Tftu
x - 2 _ y _ z-T
2 ~ 2 ~ - 1 «
x = y + 2
y = 2-2z » ( x ; y ; z ) = (4;2;0):oB(4;2;0) 2-2z + z - 2 = 0
y + z - 2 = 0
Duong thSng A B nam trong mat phSng (P) cat tai A , cat d 4 tai B '
Duong thSng AB, d ^ d j ciing chua trong (P), ngoai ra AB = ^'2'~2
khong cung phuong voi Uj = (l;2;-2), do do AB cat dj va d j
Vay A chinh la duong th5ng AB di qua B(4;2;0),c6 vecto chi phuong
OETHITHlJfSOSl
I PHAN CHUNG CHO TAT CA CAC T H I SINK
Cau 1: Cho ham so' y = - — j - c6 do thj la (C)
a) Khao sat sy bien thien va ve do thj (C) cua ham so
b) GQI A , B la 2 giao diem ciia duong thang A: y = ^x voi do thj (C)
6 Tim toa do diem M thupc duong phan giac goc phan tu thu nha't sac cho
M A + M B CO gia trj nho nha't
Cau 2: Giai phuong trinh : 2 + yj2
Cau 5: Cho hinh chop S.ABC c6 day ABC la tam giac vuong can tai
AB = BC = aN/3, khoang each tir A den mat phSng (SBC) bSng aS
Cau 4: Tim
206
Cty TNHH MTV DWH Khang Vi?t
SAB = SCB = 90" Tinh the tich khoi chop S.ABC theo a va goc giua SB voi mat phing (ABC)
Cau 6: Cho a, b la cac so' thyc thoa man a^ + b^ = 4a - 3b Tim gia trj Ion nha't
va nho nha't cua bieu thuc: P = 2a + 3b
j l PHAN RIENG T h i sinh chi dvegc chgn lam mpt trong hai phan (phan A
hole B)
A Theo chiTomg trinh chuan Cau 7.a: Trong mat phang tpa dp Oxy, cho tam giac ABC vuong tai A Dinh B(1;1), duong thing AC c6 phuong trinh: 4x + 3 y - 3 2 = 0, tren tia BC lay diem M sao cho BC.BM = 75 Tim dinh C bie't ban kinh ciia duong tron ngoai tie'p tam giac A M C bang
Cau 8.a: Trong mat phang tpa dp Oxyz, cho duong th3ng ( d ) : ^^=.X_^=£1?
va (P): - X + y + 2z + 5 = 0 Viet phuong trinh duong thiing d ' nSm trong mp
(P) dong thoi each d mpt khoang bang yjli
Cau 9.a: Cho so phuc z thoa man ding thuc 2z + i.z =
z + 2iz
1 + i Hay tinh gia trj ciia bieu thuc A =
B.Theo chUorng trinh nang cao Cau 7.b: Trong mat phSng tpa dp Oxy, cho AABC Bie't tpa dp diem A (2;-3) va B(3;-2), di^n tich tam giac AABC la — va trpng tam G ciia tam giac thupc (Juong thing A : 3 x - y - 8 = 0 Tim tpa dp diem C
x = - l + 3t Cau 8.b: Trong mat phang tpa dp Oxyz, cho duong thang d : y = 2 - 2t va hai
[z = 2 + 2t
<JiemA(l;2;-l), B(7;-2;3) Tun diem I thupc duong thing d sao cho l A + IB
•^ho nha't
b: Co ba hpp dung 5 vien bi trong do hpp thu nha't c6 1 bi trang, 4 bi
^^n; hpp thu hai c6 2 bi trSng, 3 bi den; hop thu ba c6 3 bi tring, 2 bi den Chpn
'^gau nhien mpt hpp roi tu hpp do lay ngau nhien ra 3 bi Tinh xac suat dupe ca
^biden
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Trang 14Tuyen chgn &• Giai thiju dethi Todn h<?c - Nguyen Phii Khdnh , Nguyen Tat Thu
H(/(}NGDANGIAI
I PHAN C H U N G CHO T A T CA CAC THI SINH
Cau 1:
a) Danh cho ban dpc
b) Tpa dp A, B la nghi^m ciia phuong trinh:
A, B nam ve cimg phia doi v o i duong phan giac d: x - y - 0
GQI A'(a;b) la diem doi xung ciia A qua d nen c6:
Phuong trinh tham so cua A'B la :
K h i do M la giao diem cua A'B va d
Bie'n doi p h u o n g trinh t h u nhat: (x + y)"* - (x + y ) - 3xy (x + y)^ -1 = 0
3 2
X + y - 1 = 0 v i (x + y ) + (x + y) + x + y - 3xy (x + y ) - 3xy
= x^ + y"' + x^ + y^ - xy + X + y > 0 Phuong trinh t h u hai, tro thanh:
Trang 15Dat a = t b , ( t ^ O ) thi P = ^ i i l ^ , xet ham so i{t) = ^ ^ ^ ^ , v o i
I I PHAN R I E N G T h i s i n h chi dupe chpn lam mpt t r o n g h a i p h a n (phan A
hoac B)
A Theo chUorng trinh chuan
Cau 7.a: Toa dp d i n h A(5; 4) Goi E la giao diem cua d u o n g tron ngoai tiep cua
tarn giac A M C v a i BA thi ta c6 BA.BE = BM.BC = 75 (vi M nSm tren tia BC), t i m
dupe toa dp cua E la E(13; lO) Tarn giac AEC v u o n g tai A nen C la giao cua
d u o n g tron tarn E, ban k i n h r = 575 v o i d u o n g thang A C Toa dp cua C la
4x + 3 y- 3 2 = 0
( x - i 3 f + ( y - 1 0 f = ( 5 7 5 ) '
CauS.a: ( d ) d i qua M ( 2 ; 3 ; - 3 ) , c6 veetochi p h u o n g u = ( 4 ; 2 ; l )
Xet d u o n g thSng (d') qua M , (d') nam trong (P) va (d') 1 ( d )
B Theo chUorng trinh nang cao
Cau 7.b: Gpi M la t r u n g diem AB => M (5 _ 5